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A249548 Coefficients of reduced partition polynomials of A134264 for computing Lagrange compositional inversion.

%I A249548 #65 Feb 19 2024 01:49:19
%S A249548 1,0,1,1,1,2,1,5,1,6,3,5,1,7,7,21,1,8,8,4,28,28,14,1,9,9,9,36,72,12,
%T A249548 84,1,10,10,10,5,45,90,45,45,120,180,42,1,11,11,11,11,55,110,110,55,
%U A249548 55,165,495,165,330,1,12,12,12,12,6,66,132,132,66,66,132,22,220,660,330,660,55,495,990,132
%N A249548 Coefficients of reduced partition polynomials of A134264 for computing Lagrange compositional inversion.
%C A249548 Coefficients of reduced partition polynomials of A134264 for computing the complete partition polynomials for the Lagrange compositional inversion of A134264 (see Oct 2014 comment by Copeland there). Umbrally,
%C A249548 e^(x*t) * exp[Prt(.;1,0,h_2,..) * t] = exp[Prt(.;1,x,h_2,..) * t], where Prt(n;1,0,h_2,..,h_n) are the reduced (h_0 = 1 and h_1 = 0) partition polynomials of the complete polynomials Prt(n;h_0,h_1,h_2,..,h_n) of A134264.
%C A249548 Partitions are given in the order of those on page 831 of Abramowitz and Stegun. Formulas for the coefficients of the partitions are given in A134264.
%C A249548 Row sums are the Motzkin sums or Riordan numbers A005043. - _Tom Copeland_, Nov 09 2014
%C A249548 From _Tom Copeland_, Jul 03 2018: (Start)
%C A249548 The matrix and operator formalism for Sheffer Appell sequences leads to the following relations with D = d/dh_1.
%C A249548 Exp[Prt(.;1,0,h_2,..) * D] (h_1)^n = [h_1 + Prt(.;1,0,h_2,...)]^n = Prt(n;1,h_1,h_2,...), the partition polynomials of A134264 for g(t)/t with h_0 = 1.
%C A249548 For the umbral compositional inverses described below,
%C A249548 Exp[UPrt(.;1,0,h_2,..) * D] (h_1)^n = [h_1 + UPrt(.;1,0,h_2,...)]^n = UPrt(n;1,h_1,h_2,...).
%C A249548 The respective e.g.f.s are multiplicative inverses; that is, exp[Prt(.;1,0,h_2,..) * t] = 1/exp[UPrt(.;1,0,h_2,..) * t], so the formalism of A133314 applies.
%C A249548 The raising operator R such that R Prt(n;1,h_1,h_2,...) = Prt(n+1;1,h_1,h_2,...) is R = exp[Prt(.;1,0,h_2,...)*D] h_1 exp[UPrt(.;1,0,h_2,..)*D] since R Prt(n+1;1,h_1,h_2,...) = exp[Prt(.;1,0,h_2,...)*D] h_1 (h_1)^n = Prt(n+1;h_1,h_2,...) from the definition of the umbral compositional inverse. This may also be expressed as R = h_1 + d/dD log[exp[Prt(.;1,0,h_2,...) * D]], so, using A127671, R = h_1 + h_2 D + h_3 D^2/2! + (h_4 - h_2^2) D^3/3! + (h_5 - 5 h_2 h_3) D^4/4! + (h_6 - 9 h_2 h_4 + 5 h_2^3 - 7 h_3^2) D^5/5! + (h_7 - 28 h_3 h_4 - 14 h_2 h_5 + 56 h_2^2 h_3) D^6/6! + ... . (End)
%H A249548 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%F A249548 From _Tom Copeland_, Nov 10 2014: (Start)
%F A249548 Terms may be computed symbolically up to order n by using an iterated derivative evaluated at t=0:
%F A249548 with g(t) = 1/{d/dt [t/(1 + 0 t + h_2 t^2 + h_3 t^3 + ... + h_n t^n)]},
%F A249548 evaluate 1/n! * [g(t) d/dt]^n t at t=0, i.e., ask a symbolic math app for the first term in a series expansion of this iterated derivative, to obtain Prt(n-1).
%F A249548 Alternatively, the explicit formula in A134264 for the numerical coefficients of each partition can be used. (End)
%F A249548 From _Tom Copeland_, Nov 12 2014: (Start)
%F A249548 The first few partitions polynomials formed by taking the reciprocal of the e.g.f. of this entry's e.g.f. (cf. A133314) are
%F A249548 UPrt(0) = 1
%F A249548 UPrt(1;1,0) = 0
%F A249548 UPrt(2;1,0,h_2) = -1 h_2
%F A249548 UPrt(3;1,0,h_2,h_3) = -1 h_3
%F A249548 UPrt(4;1,0,h_2,..,h_4) = -1 h_4 + 4 (h_2)^2
%F A249548 UPrt(5;1,0,h_2,..,h_5) = -1 h_5 + 15 h_2 h_3
%F A249548 UPrt(6;1,0,h_2,..,h_6) = -1 h_6 + 24 h_2 h_4 + 17 (h_3)^2 + -35 (h_2)^3
%F A249548 ...
%F A249548 Therefore, umbrally, [Prt(.;1,0,h_2,...) + UPrt(.;1,0,h_2,...)]^n = 0 for n>0 and unity for n=0.
%F A249548 Example of the umbral operation:
%F A249548 (a. +  b.)^2 = a.^2 + 2 a.* b. + b.^2 = a_2 + 2 a_1 * b_1 + b_2.
%F A249548 This implies that the umbral compositional inverses (see below) of the partition polynomials of the Lagrange inversion formula (LIF) of A134264 with h_0=1 are given by UPrt(n;1,h_1,h_2,...,h_n) = [UPrt(.;1,0,h_2,...,h_n) + h_1]^n, so that the sequence of polynomials UPrt(n;1,h_1,h_2,...,h_n) is an Appell sequence in the indeterminate h_1. So, if one calculates UPrt(n;1,h_1,...,h_n), the lower order UPrt(n-1;1,h_1,...,h_(n-1)) can be found by taking the derivative w.r.t. h_1 and dividing by n. Same applies for Prt(n;1,h_1,h_2,...,h_n).
%F A249548 This connects the combinatorics of the permutohedra through A133314 and A049019, or their duals, to the noncrossing partitions, Dyck lattice paths, etc. that are isomorphic with the LIF of A134264.
%F A249548 An Appell sequence P(.,x) with the e.g.f. e^(x*t)/w(t) possesses an umbral inverse sequence UP(.,x) with the e.g.f. w(t)e^(x*t), i.e., polynomials such that P(n,UP(.,x))= x^n = UP(n,P(.,x)) through umbral substitution, as in the binomial example. The Bernoulli polynomials with w(t) = t/(e^t - 1) are a good example with the umbral compositional inverse sequence UP(n,x) = [(x+1)^(n+1)-x^(n+1)] / (n+1) (cf. A074909 and A135278). (End)
%e A249548 Prt(0) = 1
%e A249548 Prt(1;1,0) = 0
%e A249548 Prt(2;1,0,h_2) = 1 h_2
%e A249548 Prt(3;1,0,h_2,h_3) = 1 h_3
%e A249548 Prt(4;1,0,h_2,..,h_4) = 1 h_4 + 2 (h_2)^2
%e A249548 Prt(5;1,0,h_2,..,h_5) = 1 h_5 + 5 h_2 h_3
%e A249548 Prt(6;1,0,h_2,..,h_6) = 1 h_6 + 6 h_2 h_4 + 3 (h_3)^2 + 5 (h_2)^3
%e A249548 Prt(7;1,0,h_2,..,h_7) = 1 h_7 + 7 h_3 h_4 + 7 h_2 h_5 + 21 h_3 (h_2)^2
%e A249548 ...
%e A249548 ------------
%e A249548 With h_n denoted by (n'), the first seven partition polynomials of A134264 with h_0=1 are given by the first seven coefficients of the truncated Taylor series expansion of the Euler binomial transform
%e A249548 e^[(1') * t] * {1 + 1 (2') *  t^2/2! + 1 (3') *  t^3/3! + [1 (4') + 2 (2')^2] *  t^4/4! + [1 (5') + 5 (2')(3')] *  t^5/5! + [1 (6') + 6 (2')(4') + 3 (3')^2 + 5 (2')^3] *  t^6/6!}, giving the truncated expansion
%e A249548 1 + 1 (1') * t + [1 (2') + 1 (1')^2] * t^2/2! + ... + [1 (6') + 6 (1')(5') + 6 (2')(4') + 3 (3')^2 + 15 (1')^2(4') + 30 (1')(2')(3') + 5 (2')^3 + 20 (1')^3(3') + 30 (1')^2(2')^2 + 15 (1')^4(2') + 1 (1')^6] * t^6/6!.
%e A249548 Extending the number of reduced partition polynomials of the transform allows for further complete polynomials of A134264 to be computed.
%t A249548 rows[n_] := {{1}, {0}}~Join~Module[
%t A249548     {g = 1 / D[t / (1 + Sum[h[k] t^k, {k, 2, n}] + O[t]^(n+1)), t], p = t, r},
%t A249548     r = Reap[Do[p = g D[p, t]/k; Sow[Expand[Normal@p /. {t -> 0}]], {k, n+1}]][[2, 1, 2 ;;]];
%t A249548     Table[Coefficient[r[[k]], Product[h[t], {t, p}]], {k, 2, n}, {p, Sort[Sort /@ IntegerPartitions[k, k, Range[2, k]]]}]];
%t A249548 rows[12] // Flatten (* _Andrey Zabolotskiy_, Feb 18 2024 *)
%Y A249548 Cf. A134264, A005043, A133314, A049019, A074909, A135278.
%Y A249548 Cf. A127671.
%Y A249548 Rows lengths are given by A002865 (except for row 1).
%K A249548 nonn,tabf
%O A249548 0,6
%A A249548 _Tom Copeland_, Oct 31 2014
%E A249548 Formula for Prt(7,..) and a(12)-a(15) added by _Tom Copeland_, Jul 22 2016
%E A249548 Rows 8-12 added by _Andrey Zabolotskiy_, Feb 18 2024